Hermeneutics: Applying Mathematical Logic To The Quran (Part 1)

The Quran is a sacred text that has been the subject of interpretation and analysis for centuries. While classical methods of Quranic exegesis have relied heavily on linguistic conventions and presupposed meanings, the advent of modern symbolic logic offers a new and powerful framework for deriving meaning from the Quran's words. By applying the principles and rules of symbolic logic, which are grounded in the precise language of mathematics, we can analyze Quranic terms in a more systematic and rigorous way, uncovering new insights and connections that traditional methods may overlook. This essay will explore how symbolic logic can be used as a tool for Quranic interpretation, focusing on the application of key logical rules such as non-contradiction, excluded middle, identity, and sufficient reason.

Section 1: The Shift from Classical to Symbolic Logic

The development of symbolic logic marked a significant shift in the way we analyze and interpret language. By representing propositions using clear, unambiguous mathematical notations, symbolic logic allows us to study their structure and relationships with a new level of precision and rigor. This has led to groundbreaking advances in fields like mathematics, philosophy, and computer science. As we will see, applying symbolic logic to the Quran can similarly yield new and valuable insights into the meaning of its words.

What we see in classical logic is merely the 'outline', the 'silhouette' of that science, and it's what some of us (as I surely did), thought most of logic consisted of (which is essentially classical logic), until I came upon 'logicism', and the work of Bertrand Russell, Frege, Wittgenstein, Boole and other (relatively) modern logicians. Here logic is treated by people who also dealt with set theory, arithmetic, algebra, geometry, analysis etc.., and thus the form of the logical 'proposition' which used to be expressed with - and as a consequence, complicated by - language, was embodied in the form of the symbolic formula, which among other benefits, does away with the fetters of linguistic convention, leaving nothing unexamined. Symbolic logic involves construction, that is, 'showing' or 'actually expressing' the proposition through the arrangement of symbol systems and notations.

This by no means says that symbolic logic is 'non-linguistic' (in the colloquial sense of language), or that 'language' broadly speaking is 'informal' (in that logical sense of 'formality'), both are nested in the same 'expressive' 'élan' (to use the Bergsonian term). The difference is that symbolic logic is, let's say, more 'deliberate' with respect to the principles of logic. It wants to express just that 'axiom' or 'formality' which, by the way, it perceives in language. It 'sees it', 'grasps it', then 'reformulates' in a way that can be 'manipulated' - and this is the chief purpose of formal logic - use it in areas 'outside' the original domain of inquiry.

The pinnacle of abstraction is reached in mathematics. If we inquire about what characterizes mathematics and what makes it so valuable, everyone would agree that it resembles language, but with a crucial difference: it enables us to articulate concepts that are beyond the scope of ordinary language, such as ideas related to measurement, magnitude, and geometric relationships. This is not to suggest that these concepts cannot be expressed using linguistic propositions. However, when we attempt to do so, the precision inherent in mathematical symbolic notation is not completely lost, but rather substituted with less exact generalizations. It is comparable to a specialist trying to explain a highly intricate concept to a non-expert audience.

This dialectical relationship, where each faculty lacks the very thing that the other seems particularly suited to provide, is reminiscent of Heidegger's distinction between the "ready-at-hand" and the "present-at-hand." Mathematics deals solely with the "present-at-hand," leaving the "ready-to-hand" to the philosopher, whose attempts at rigor are hindered by the inherent generality of ordinary language.

However, mathematicians face a similar challenge: their rigor aims for generality, but they can only express it in the coded language of mathematics. Thus, both mathematicians and philosophers grapple with the limitations of their respective tools in capturing the full scope of their insights. The fact of the matter is, the 'success' of the mathematician rides as much on the precision of his mathematical techniques and the 'consistency' of his proofs, as on their 'generalizability' or 'expressibility' in general terms.

It goes back to the original point, 'reformulation'; or, as we reformulated it in this context 'generalizability' (the degree to which research findings can be applied to other situations).

These two tendencies are synthesized in logic, as in; we can represent the 'linguistic' in mathematics, and (thanks to the project of logicism) the 'mathematical' in language. It informs us that these two fields not only have a common link, but that this link is unbreakable. Can 'text' be made as poetically expressive as language, yet as deductively clear as a mathematical proposition?

This is the central question of the philosophy of language, and to a greater degree, of the 'theory of knowledge'. In the Quran, we discover that the language itself embodies this concept. However, a reader who approaches the Quran as they would any other text will not perceive this, and they would likely acknowledge this themselves. Let's conduct a small experiment to illustrate this point.

First, open the Quran to any chapter and verse, and keep the book open. Next, pick up another book from your shelf, possibly a beloved novel. Read one page from the novel, and then shift your focus to the Quran, reading a page from there as well.

Now, if I were to ask you about the difference between the two texts, your most probable response would be that it lies in the subject matter they cover. This observation suggests that the distinction between the Quran and other text is not immediately apparent, or rather, that the two texts can be read the same way.

Perhaps you're of the analytical sort and have a sharper response that says that the difference lies in the literary structure - that the Quran is didactic, instructive and legalistic, while your novel is more satirical, romantic or visual. But the fact is, we're nowhere near what makes these two texts different. Comparing their literary style or content assumes they can be approached in the same way. This approach is fundamentally flawed from the start.

"You're suggesting that the problem with conventional approaches to reading the Quran is that they don't sufficiently take into account the reader's own capacity to understand and access the text itself; and that we rely too much on given interpretations.

In the philosophical tradition of Immanuel Kant, true understanding requires 'critical subjectivity' - an active reflection on how our own mental faculties shape our interpretation. To adequately grasp the Quran's meaning, we must start by examining our own role and assumptions as readers, rather than treating the text as a passive object of analysis. Reflection is not straightforward either; it proceeds by rules; rules that when correctly followed always leads to profound insights, but there's a problem here; we need to understand what those rules are, and the order of their operation, their relations to one another, and how to properly wield them; as the example of Joseph in chapter 12, verse 4 shows.

Here, we're not comporting to what the text 'seems to be saying', but are 'deliberating' our approach. The Quranic 'word' is not 'given' in the sense that the actual meaning is that which the passage 'conveys' to me as a person immersed in the tradition, but rather 'given' in the sense that its meaning is 'potentially in the text' at large. You may liken this approach to 'induction' or 'falsification', but as this topic prioritizes the symbolic-logical approach, we want something a little more technical than that. You'll find this straightforward, as you already engage in the sort of 'reasoning' that we're going to describe.

Joseph's vision of the 11 constellations is a manifestation of the intrinsic harmony and arrangement that characterises their existence. The word Ahada, when combined with Asshara, which signifies a group or class of entities sharing a common attribute conveys a sense of unity and cohesion. This combination, forming the number 11, represents a structured arrangement, where the individual elements are understood as belonging to a larger, interconnected whole. The reference to 11 constellations in this context is not merely a numerical count, but rather an expression of an underlying organisational principle. A myriad logic relationship that binds these celestial bodies together as parts of a harmonious cosmic system.

Logical analysis

The first, major one, is 'Tautology'. In propositional logic, a tautology is a formula or proposition that is true in every possible interpretation. This is what your critical analysis commences with. But note that 'true' here isn't something 'controvertible'; something you may find 'agreeable' or otherwise, it's what you must accept first before you could find it agreeable or disagreeable. It's the form of general existential 'validity'. It is true insofar as you 'have a proposition at your grasp', something to deal with, something to judge.

P ∨ ¬P

Let's consider the statement "London is the capital of Austria." Based on your knowledge of geography, you will object and assert that this claim is false. This reaction is perfectly understandable. However, from the perspective of logic, specifically the rule of tautology, the statement would be considered true. The tautological truth here does not depend on the actual geographical facts about London or Austria. Instead, it relies on the mere existence of the proposition itself.

Logic would argue that for you to even declare the statement false, you must first acknowledge it as a valid proposition. Were it not valid, it would not be comprehensible enough to be factual or otherwise, agreeable, or outrageous, it simply would not exist. In other words, you have to recognise nize that the assertion "London is the capital of Austria" has been made, regardless of its factual accuracy.

From a logical standpoint, the truth value here is not derived from the correspondence between the claim and reality, but rather from the presence of the proposition itself. The fact that someone has put forth this statement is sufficient for it to be considered logically true. So, while you are wrong in terms of geographical facts, logic would maintain that the statement is true based on the principle of tautology. It is the existence of the proposition, not its factual accuracy, that establishes its logical truth.

I infer from this that logical falsehood must be equivalent to logical truth, which would mean that the basis of falsehood here is not 'correspondence' to what we think of as reality, i.e., that London is actually a city in England, and Vienna is the capital of Austria. Reality for it is the structure of a proposition; that there is a proposition is reality. This implies that if judged from the standpoint of correspondence, we're making a reference to some other proposition to which we attach the notion of logical validity a priori, and reject the so-called false proposition on the basis of its 'inconsistency' with it, or, let's say, both are claiming the same underlying thing, but one of them is false merely because it's different from the other.

It's clear to me that logic does not deal with my original sense of what makes something false. Yes, it is as you said, because logic deals with structure and structure betrays traditional notions of truth and falsehood; because structure is relative, and relativity has to do with the order of things in any given instance, and order varies, and at any given instance, what's the case may not be, and this relativity is needed to look at things from different perspectives, and analysis requires that both the order as it is, and as it could be, be both valid. This is tautology, in a nutshell.

That there is a proposition, is reality, and that there could be such a proposition, is reality also; this makes the 'false' in the traditional sense, a 'possible truth', an approximation, a deferred statement, or something that anticipates a truth. But as you saw, tautology represents the valid and the not yet valid or what exists and what could exist, insofar as both simply 'are', and this is what the symbolic formula expresses (p v -p), that the truth, p, and the falsehood, -p, share the variable p as a common denominator.

But I see more in it; if tautology asserts the validity of 'what is' and 'what could be', or let's say, the valid and the possible as we put it, does this not mean that there's a dialectical relation between them?, and that there are two truths and two falsehoods? That 'what is' is false when 'what could be' is asserted as valid, since it is asserted as opposed to it, and vice versa?.

It clearly shows that both have to be true and false at least once, even if they're originally held to be either true or false. This difference seems to bring out the meaning of 'what could be', because, as I see it, there's at least another possible state that a proposition may assume when held to be in a particular state.

What we're describing here is the rule of absorption;

p ^ (p v q) = p p v (p ^ q) = p

it is that rule by which we preserve the 'validity' of some proposition when not immediately asserted. Absorption laws simplify logical formulas without changing their truth value, and by simplify, I mean 'defer' or 'withhold', 'peserve', ensuring equivalence, analogous to how a tautology is true under all circumstances.

Suppose we are counting apples in a fruit stall. You are using the number system (natural numbers) to count 1,2,3,4 etc... let's ask, where do the rest of the numbers go as we count each? When you count 3, where are the rest of the numbers? Certainly, they are 'somewhere', but where? The rest of the numbers (4, 5, 6, ...) are always present within the mathematical system you're using, just not 'asserted', but they exist, and as such fall within the range of tautological validity.

In conjunction with tautology, we find the rule of absorption; this is the linchpin of validity. If you examine the formula, you will think initially that there is one truth and one falsehood, but then your acquaintance of tautology will have revealed to you another dimension that shows two truths; namely, that p is true and that not -p, i.e., the false p is true as well!; because, you learn that in order to say of not p that it's false, it must be true that it's false. Thus the false must at some level be truly false.

Absorption is the flipped side of this; there's a state in which p is not yet true; where it's possible or anticipated, but not yet explicitly noted as true. This is evident from the fact that we have a scheme to begin with, and since we do, we infer that this scheme was possible. This is a first sense of falsehood with which the fundamental state of bare validity is contrasted. These two are part and parcel of each other, where there's 'anything' there was the possibility of its being, regardless of what it is, as this is the basic duality of the mind. But an important distinction to be made is between truth and falsehood as such, and the assertion that something is true and/or false.

The object/concept & The Assertion (Object language & Metalanguage)

Object 'propositions' 'Truth/False' p/-p

Metalanguage 'propositions'. p is Truth -p is False

The assertion ‘that p is true and not p is false’, are  assertions ‘about p and not p’, they are of a different order than the former, but which are necessary for their completion, and for the notions of truth and falsehood to arise in the first place. This means that we have two orders of truth, and two orders of falsehood, and what the rule of tautology states is the validity of assertions at all levels; because, as a rule, since it has to do with the truth at large even the falseness of some truth, we infer that it itself arises from the condition of falsehood (the 'Penumbral', the 'null'), because what’s asserted must be asserted at the expense of what’s not asserted, which in this case is the ‘possibility’ of said assertion. 

Consider how a proposition points to itself - a kind of self-reference. While this might seem paradoxical, particularly given Russell's establishment of a whole branch of logic dealing with this problem (The Doctrine Of Types found in the Principia Mathematica 1910-13, and the Principles of Mathematics 1903 - appendix B), it actually isn't paradoxical at all.

To Emphasize, the assertion about p or not p, and p and not p as such are not equivalent; the former (the assertion), is a limiting case of the objects. This necessitates their belonging to different orders, which is precisely Russell's solution; the assertion about p is a limiting case of the object p, inasmuch as an instance of the number 2 is a limiting case of the universal number 2 that can exist in a different instance. This is known in propositional logic, which is why we have q and not q. This makes q and ¬q 'metalogical' signs.

1) q is the assertion of the truth of p,

2) ¬q the assertion of the falsehood of not p.

A crucial observation emerges here: what manifests are rules of inference. Particularly modus tollens and modus ponens, as these rules seem to place p in conjunction with q, in the case of modus ponens, and ¬q in conjunction with -p in modus tollens, which shows that they are rules which govern the relation between these two orders. They describe 'trans-ordinal' relations.

Furthermore, we maintain the possibility to refer directly and laterally to the truth of any of these propositions a priori (-p), which creates a disjunctive syllogism → insofar as possibility -p is anticipates the assertion of some instance of truth (or falsehood) → q, and q generally applies to all that is 'assertable'. This is consistent so long as:

A) We maintain that q is a limiting case of truth,

B) We maintain that Truth is merely what's actually asserted;

Thus q also applies to falsehood (assertorically), if we hold that the assertion of possibility is a still a true assertion. However, this cannot be stated without involving the hypothetical syllogism which treats any possible truth -q as an original truth p; in other words, the object of any reference would be a reference already given within the system, we never 'conjure' up anything out of nowhere, what's asserted, is something 'assertable', that is to say, 'given' to exist before the assertion.

What am I yapping about?, let's simplify this...

• When we say something is possible (let's call this -p), we're already making a true statement about that possibility; this mode or 'manner of speaking' is what I call 'assertoric' (it's only possible to say of falsehood that it's truely false' as an 'assertion')

• This creates what's called a "disjunctive syllogism" - basically, a way of reasoning that says "if this is possible, then we can make true or false statements about it"

• When we say something is true (q), we're really just pointing out what's already there, a hypothetical syllogism:

  
  

  if (p → q) and (q → r), then (p → r). it's the old syllogism that says that because Socrates is a man, and all men are mortal, Socrates must be mortal. Think of 'all men are mortal' as being embodied in (q → r). This makes it an 'assertoric' proposition, the 'argument' that's embedded prior to the conclusion that was the original 'point'.

A significant transformation occurs in the definition of truth and falsehood when two different orders of it are posited;

When we think about truth and falsehood on two different levels, something interesting happens. Think of it like this: When we say something is "not true" (-p), we're actually making a true statement (q) about its falseness. And interestingly, whenever we make a true statement about something being false, we're showing that falsehood itself has a kind of validity. At the same time, our ability to say something is false (-q) → i.e 'to validate falsehood', is what allows us to make true statements in the first place, because truths are contrasted with falsehood, and arise from them: and in order to 'arise' from them, they (falsehood) must be valid → precisely what we accomplished by the asserting/validating falsehood in the first place. There's no 'linearity' here, the rules are interdependent and interrelated 'apriori'.

• There are two truths; a higher order (assertoric truth), and a lower order ( truth)

• There are two falsehoods; a higher order (assertoric falsehood), and a lower order (Falsehood)

• You validate a falsehood only because you 'falsify' a truth in the same context

• You falsify a truth only because you validate a falsehood

This pattern is known/used in algebra, particularly in topology and set theory. Venn diagrams capture them

Thus this order explains the relation between truth and falsehood and how each qualifies the other, allowing for an understanding of the relation of truth to falsehood as a validation, and of falsehood to truth as 'potentiation' or 'anticipation'. In other terms: Truth becomes a validating principle when it occupies that position formally assigned to falsehood, and likewise, falsehood becomes a 'potentiating', intuitional principle when it occupies that place formally assigned to truth. More simply put: truth presupposes the possibility of this truth, inasmuch as the possibility is a possibility of truth.

Moreover, it becomes apparent that the rules of tautology and absorption emerge as necessary interpretive schema when non-contradiction is thought in light of the rules of inference. These four rules coordinate the position of each component, and by extension, every possible derivable combination from the two vaues .of truth and falsehood (here's a list).

(p → p)    (p → q)  (-p → p)  (-p → q)   (-p → -q)   

(q → q)    (q → p)  (-q → q)  (-q → p)   (-q → -p)

(p → -p)   (p → -q)   (-q → q)   (-q → -p)

All these combinations fall under non-contradiction; even if we expanded them into tetradic (or more) groups like:

(p → -q) ^ (or v) (-p → q)

In this case, you can read this as expressing how a resultant is derived from an operation. Here, a 'truth' implies a higher order falsehood (p → -q), meaning we only obtain a truth because it was possible to: Truth was a possibility, that is to say a 'potential truth'.

Now add to this proposition, the next

(-p → q): that a falsehood (-p) implies a higher order truth, saying that -p, though false (originally), is a valid assertion. Combining will give us a meaningful proposition. It shows how -p, falsehood, the same falsehood on which the implicitness of the truth rests, is a 'valid' assertion. It affirms the basis on which we have an 'implicit truth' in the first place: it validates that condition → implicitness' (essentially falsehood), that is the basis of the truth with which we started our analysis. in short, we assert the possibility of truth (validity) (p → -q), then we assert the validity of 'possibility' (falsehood)

This is, admittedly, a crude way of positing rules of replacement which explain the different ways that the two orders, whose basic equivalence and relation is established by the rules of inference, relate to each other.

Follow us over to part 2.